Bug or unfortunate feature?

[David Guichard]

When is a linear system consistent?

This isn't helpful:

a,b,c=var('a b c')

A=matrix([[-4,5,9,a],[1, -2, 1, b],[-2,4,-2,c]])
A.echelon_form()

A.rref()

[    1     0 -23/3     0]
[    0     1 -13/3     0]
[    0     0     0     1]

[    1     0 -23/3     0]
[    0     1 -13/3     0]
[    0     0     0     1]

But this works, so I know how to get what I want:

R.<a,b,c>=QQ[]
A=matrix(R,[[-4,5,9,a],[1, -2, 1, b],[-2,4,-2,c]])
A.echelon_form()

[ 1 0 -23/3   -2/3*a - 5/3*b]
[ 0 1 -13/3   -1/3*a - 4/3*b]
[ 0 0     0             2*b + c]

So [a,b,c] is in the span of the columns iff 2b+c==0.

Somewhat oddly, rref doesn't work here:

A.rref()

[    1     0 -23/3     0]
[    0     1 -13/3     0]
[    0     0     0     1]

[David Joyner]

This is because the CAS used assumes  2*b + c is non-zero.
I would call it an "unfortunate feature", except that it (this "feature")

gives teachers of linear algebra examples for tests and quizzes that

can't be correctly solved using a CAS, and have to be solved by hand.

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[Thierry]

Hi,

I would say that there is a bug in the 'default' algorithm. Note that
changing the algorithm leads to:

sage: A.echelon_form(algorithm='partial_pivoting')
[ 1 0 -23/3 -2/3*a + 5/6*c]
[ 0 1 -13/3 -1/3*a + 2/3*c]
[ 0 0 0 b + 1/2*c]

Ciao,
Thierry

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[Nils Bruin]

On Friday, September 6, 2019 at 1:19:48 AM UTC-7, Thierry (sage-googlesucks@xxx) wrote:

I would say that there is a bug in the 'default' algorithm. Note that
changing the algorithm leads to:

sage: A.echelon_form(algorithm='partial_pivoting')
[             1              0          -23/3 -2/3*a + 5/6*c]
[             0              1          -13/3 -1/3*a + 2/3*c]
[             0              0              0      b + 1/2*c]

While that may be a useful answer for this particular problem, the real issue is that linear algebra over rings that are not fields is more complicated. Over PIDs one can use Hermite Normal Form as a substitute for reduced row echelon form, in general this gets more complicated. Even unique factorization domains are not so easy to handle.

For instance, over QQ[a,b,c,d], how would you echelonize the following matrix?

[ a b ]

[ c d ]

There are algorithms to compute with modules over polynomial rings (i.e., do linear algebra over polynomial rings), but they require more complicated methods (basically groebner bases) and the answers one gets are more complicated too, because these modules are not just determined by their free rank (the analogue of dimension).

So for the original problem I'd go with: "unfortunate mathematical reality" rather than "unfortunate feature".